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[4] viXra:2604.0101 [pdf] submitted on 2026-04-25 22:42:36
Authors: Theo Adebayo
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper develops the ordered-structure core of the Theory of Residual Cancellation (TRC).The main idea is that, in a suitable ordered setting, the common part of two positive elements can be recovered from ordered difference and positive-part structure rather than assumed independently. Let G be an ordered abelian group equipped with a positive-part operation u → u+, and define u− := (−u)+. For x,y ∈ G, define the TRC common-part candidatem(x,y) := x−(x−y)+. Under a positive/negative-part decomposition axiom, this operation is symmetric and yields a two-sided residual/common decomposition. Under the additional assumption that the positive part map is monotone, the matched-content operation becomes monotone, maximal among common lower bounds, and equal to the meet on positive pairs. This gives an axiom-separated theorem ladder: one axiom governs the algebraic decomposition layer, while the second upgrades the decomposition into genuine order-theoretic meet recovery. The result identifies the abstract core of TRC as a compatibility principle between difference, positive part, and common part.
Category: Algebra
[3] viXra:2604.0100 [pdf] submitted on 2026-04-26 18:40:33
Authors: Sergey Y. Kotkovsky
Comments: 33 Pages. In Russian
As a base for creating nonlinear algebra, we use vectors — mathematical objects with some predefined general properties, but without defining these objects in terms of numbers or numerical matrices. Next, we build our algebra based on the vector multiplication operation. Our approach allows us to obtain new and more generalized conceptions of vectors, scalars and related objects of a mixed scalar-vector type — generalized quaternions. We propose a fundamentally new perspective on such familiar concepts as space, vectors, quaternions, complexity, parallelism, orthogonality, and dimension. Within the framework of new nonlinear algebra, geometric concepts such as parallelism and orthogonality acquire the operator meaning of vector commutativity and anticommutativity. The essence of the transition from linear to nonlinear representations lies in the transition from static geometric representations to operational ones. Vector cycles, which are triples of vectors cyclically connected to each other, occupy a special place in our algebraic system. The axiomatic framework we have constructed allows us to prove a number of statements important for the further development of the theory of nonlinear spaces.
Category: Algebra
[2] viXra:2604.0075 [pdf] submitted on 2026-04-20 22:17:01
Authors: Jose Risomar Sousa
Comments: 4 Pages.
I present a method to solve the general cubic polynomial equation based on six years of research that started when, in the fifth grade, I first learned of Bhaskara's formula for the quadratic equation. I was fascinated by Bhaskara's formula and naively thought I could replicate his method for the third degree equation, but only succeeded after countless failed attempts. The solution involves a simple transformation to form a cube and which, by chance, happens to reduce the degree of the equation from three to two (which seems to be the case of all polynomial equations that admit solutions by means of radicals).
Category: Algebra
[1] viXra:2604.0038 [pdf] submitted on 2026-04-11 01:43:47
Authors: Harish Chandra Rajpoot
Comments: 38 Pages. (Note by viXra Admin: The references are not listed in a standard/complete manner)
A generalized series-based formulation is developed to determine the hierarchical rank of any given linear permutation selected from a set of all possible linear permutations arranged according to a predefined order of priority of elements like digits, letters, and all other objects. The proposed model applies to permutations of words, numbers, and other discrete objects, enabling systematic identification of their positions in an ordered sequence. The formulation is expressed as a finite series in which each term corresponds to a specific element of the permutation. It applies to sets of distinguishable objects characterized by identifiable properties such as shape, size, colour, or surface pattern, assuming that all elements are equally likely to occupy any position in the arrangement without replacement. The model introduces three parameters, Formerity (F), Permuty (P), and Similarity (S), which collectively define the structure of the series. These parameters depend on the preceding elements, the permutations of successive elements, and the repetition characteristics within the arrangement. Notably, the number of terms in the series is equal to the number of elements in the permutation. This generalized formulation provides a structured and scalable approach for analyzing and ranking linear permutations in a wide range of combinatorial contexts.
Category: Algebra
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